Study of Optimization Problems by Quantum Annealing

We introduce quantum fluctuations into the simulated annealing process of optimization problems, aiming at faster convergence to the optimal state. Quantum fluctuations cause transitions between states and thus play the same role as thermal fluctuations in the conventional approach. The idea is tested by the two models, the transverse Ising model and the traveling salesman problem. Adding the transverse field to the Ising model is a simple way to introduce quantum fluctuations. The strength of the transverse field is controlled as a function of time similarly to the temperature in the conventional method. The goal is to find the ground state of the diagonal part of the Hamiltonian with high accuracy as quickly as possible. We also consider the traveling salesman problem. This model can be described by the Ising spin, so that we also apply the same technique to the transverse Ising model. We solve the time-dependent Schrödinger equation numerically for small size systems with various types of exchange interactions of the Ising model. Comparison with the results of the corresponding classical (thermal) method reveals that the quantum method leads to the ground state with much larger probability in almost all cases if we use the same annealing schedule of the control parameters. We check the case of large-size systems by using the quantum Monte Carlo method. The simulation supports the results of the small-size systems, while the dynamics of the Schrödinger equation and the quantum Monte Carlo method are not the same. We find that the simulated annealing by quantum fluctuations has a better performance than the conventional method for the ground state search of the Ising-spin systems. The calculation of the traveling salesman problem is also performed as